Convergence of the all-time supremum of a L\'evy process in the heavy-traffic regime

In this paper we derive a technique of obtaining limit theorems for suprema of L\'evy processes from their random walk counterparts. For each $a>0$, let $\{Y^{(a)}_n:n\ge 1\}$ be a sequence of independent and identically distributed random variables and $\{X^{(a)}_t:t\ge 0\}$ be a L\'evy processes such that $X_1^{(a)}\stackrel{d}{=} Y_1^{(a)}$, $\mathbb E X_1^{(a)}<0$ and $\mathbb E X_1^{(a)}\uparrow0$ as $a\downarrow0$. Let $S^{(a)}_n=\sum_{k=1}^n Y^{(a)}_k$. Then, under some mild assumptions, $\Delta(a)\max_{n\ge 0} S_n^{(a)}\stackrel{d}{\to} R\iff\Delta(a)\sup_{t\ge 0} X^{(a)}_t\stackrel{d}{\to} R$, for some random variable $R$ and some function $\Delta(\cdot)$. We utilize this result to present a number of limit theorems for suprema of L\'evy processes in the heavy-traffic regime

Large closed queueing networks in semi-Markov environment and its application

The paper studies closed queueing networks containing a server station and $k$ client stations. The server station is an infinite server queueing system, and client stations are single-server queueing systems with autonomous service, i.e. every client station serves customers (units) only at random instants generated by a strictly stationary and ergodic sequence of random variables. The total number of units in the network is $N$. The expected times between departures in client stations are $(N\mu_j)^{-1}$. After a service completion in the server station, a unit is transmitted to the $j$th client station with probability $p_{j}$ $(j=1,2,...,k)$, and being processed in the $j$th client station, the unit returns to the server station. The network is assumed to be in a semi-Markov environment. A semi-Markov environment is defined by a finite or countable infinite Markov chain and by sequences of independent and identically distributed random variables. Then the routing probabilities $p_{j}$ $(j=1,2,...,k)$ and transmission rates (which are expressed via parameters of the network) depend on a Markov state of the environment. The paper studies the queue-length processes in client stations of this network and is aimed to the analysis of performance measures associated with this network. The questions risen in this paper have immediate relation to quality control of complex telecommunication networks, and the obtained results are expected to lead to the solutions to many practical problems of this area of research.Comment: 35 pages, 1 figure, 12pt, accepted: Acta Appl. Mat